CHAPTER VI.

Of the Discharge from compound or divided Reservoirs.

The author observes from Buat, that the discharge through an orifice between two reservoirs, below the surface, is the same as if the water ran into the open air. Hence he calculates the discharge when the water has to pass through several orifices in the sides of as many reservoirs open above. In such cases, where the orifices are small, the velocity in each may be considered as generated by the difference of the heights in the two contiguous reservoirs, and the square root of the difference will, therefore, represent the velocity; which must be in the several orifices, inversely as their respective areas; so that we may calculate from hence the heights in the different reservoirs when the orifices are given. Mr. Eytelwein then considers the case of a lock which is filled from a canal of an invariable height, and determines the time required, by comparing it with that of a vessel emptying itself by the pressure of the water that it contains, observing that the motion is retarded in both cases in a similar manner, and he finds the calculation agree sufficiently well with experiments made on a large scale. The motion of water through different compartments of a closed cavity is also determined.

CHAPTER VII.

Of the Motion of Water in Rivers.

The simple theorem by which the velocity of a river is determined, appears to be the most valuable of Mr. Eytelwein's improvements; although the reasoning from which it is deduced is somewhat exceptionable. The friction is nearly as the square of the velocity; not because a number of particles proportional to the velocity are torn asunder in a time proportionally short, for, according to the analogy of solid bodies, no more force is destroyed by friction when the motion is rapid than when slow, but because, when a body is moving in lines of a given curvature, the deflecting forces are as the squares of the velocities, and the particles of water in contact with the sides and bottom must be deflected, in consequence of the minute irregularities of the surfaces on which they slide, nearly into the same curvilinear path, whatever their velocity may be.

At any rate, we may safely set out with the hypothesis, that the principal part of the friction is as the square of the velocity. And the friction is nearly the same at all depths; for Professor Robison found that the time of the oscillation of the fluid in a bent tube was not increased by increasing the pressure against the sides, being nearly the same when the principal part of the tube was situated horizontally as when vertically. The friction will, however, vary according to the surface of the fluid which is in contact with the solid, in proportion to the whole quantity of fluid: that is, the friction for any given quantity of water will be as the surface of the bottom and sides of a river directly, and as the whole quantity of water in the river inversely; or supposing the whole quantity of water to be spread on a horizontal surface, equal to the bottom and sides, the friction is inversely

as the height at which the river would then stand, which is called the hydraulic mean depth.

Now, when a river flows with an uniform motion, and is neither accelerated nor retarded by the action of gravitation, it is obvious that the whole weight of the water must be employed in overcoming this friction; and, if the inclination vary, the relative weight, or the force that urges the particles along the inclined plane, will vary as the height of the plane when the length is given, or as the fall in a given distance; consequently, the friction, which is equal to the relative weight, must vary as the fall, and the velocity, which is as the square root of the friction, must be as the square root of the fall; and, supposing the hydraulic mean depth to be increased or diminished, the inclination remaining the same, the friction would be diminished or increased in the same ratio; and, therefore, in order to preserve its equality with the relative weight, it must be proportionally increased or diminished by increasing the square of the velocity, in the ratio of the hydraulic mean depth, or the velocity in the ratio of its square root. We may, therefore, expect that the velocities will be conjointly as the square root of the hydraulic mean depth, and of the fall in a given distance, or as a mean proportional between these two lines. Taking two English miles for a given length, we must find a mean proportional between the hydraulic mean depth and the fall in two miles, and inquire what relation this bears to the velocity in a particular case, and thence we may expect to determine it in any other. According to Mr. Eytelwein's formula, this mean proportional is 1 of the velocity in a second.

In order to examine the accuracy of this rule, we may take an example which could not have been known to Mr. Eytelwein. Mr. Watt observed, as Professor Robison informs us, in the article "River" of the Encyclopædia Britannica, that in a canal 18 feet wide above, and 7 below, and 4 feet deep, having a fall of four inches in a mile, the velocity was 17 inches in a second at the surface, 14 in the middle, and 10 at the bottom;

so that the mean velocity may be called 14 inches, or somewhat less, in a second. Now, to find the hydraulic mean depth, we must divide the area of the section, 2 (18 +7)=50, by the breadth of the bottom and length of the sloping sides added together, whence we have or 29.13 inches: and the fall in

50 20.6'

two miles being 8 inches, we have ✓ (8 × 29·13) = 15·26 for the mean proportional, of which 19 is 13.9; agreeing exactly with Mr. Watt's observation. Professor Robison has deduced from Dubuat's elaborate theorems 12.568 inches for the velocity, which is considerably less accurate.

For another example, we may take the Po, which falls 1 foot in two miles, where its mean depth is 29 feet; and its velocity is observed to be about 55 inches in a second. Our rule gives 58, which is, perhaps, as near as the degree of accuracy of the data will allow.

On the whole, we have ample reason to be satisfied with the unexpected coincidence of so simple a theorem with observation: and, in order to find the velocity of a river from its fall, or the fall from its velocity, we have only to recollect that the velocity in a second is 1 of a mean proportional between the hydraulic mean depth and the fall in two English miles. This is, however, only true of a straight river flowing through an equable channel.

For the slope of the banks of a river or canal, Mr. Eytelwein recommends that the breadth at the bottom should be of the depth, and at the surface 10: the banks will then be in general capable of retaining their form.* The area of such a section is twice the square of the depth, and the hydraulic mean

* When a canal has the proportions described in the text, the slope of the bank within the water makes an angle of 37° with the horizon, or of 4 to 3; the slope usually employed in this country is 3 to 2, making an angle of about 34°; and the relation between the breadth and depth varies considerably according to the nature of the traffic, and of the canal-boats.-ED.

depth of the actual depth. He then investigates the discharge of a canal of which the bottom is horizontal. The velocity appears, in this case, to be somewhat greater than in a similar canal of which the bottom is parallel to the surface.

The author remarks that the velocity is greater near the concave than the convex side of a flexure a circumstance probably occasioned by the centrifugal force accumulating the water on that side.* No general rule can be given for the decrease of the velocity in going downwards: but sometimes the maximum appears to be a little below the surface. In the Arno the velocities are, at 2 feet below the surface, 39 inches; at 4, 38; at 8, 37; at 16, 33; at 17, 31. In the Rhine, at 1 foot, 58 inches; at 5, 56; at 10, 52; at 15, 43. As an approximation to the mean velocity, the author directs us to deduct from the superficial velocity for every foot of the whole depth. For instance, if the depth were 13 feet, and the superficial velocity 5 feet, to take 4 as the average velocity of the whole river. This can only, however, be true in large rivers; for, in the canal measured by Mr. Watt, the superficial velocity must be diminished nearly for a depth of only 4 feet. And we may, in general, come quite as near to the mean velocity by taking of the superficial velocity; although this may still differ materially from the true medium. But, comparing this with the former theorem for the velocity, which gives a result oftener above than below the truth, we may bring them both into a form easily recollected, thus:

The superficial velocity of a river is nearly a mean proportional between the hydraulic mean depth, and the fall in two miles; and the mean velocity of the whole water is, still more nearly nine-tenths of this mean proportional.+

* When the direction of a current is changed by a bend in its channel, the portion of the stream most distant from the centre of curvature is sensibly of a. higher level than that which is nearest to the centre.-ED.

The inclination of the surface of rivers is of more importance than is gene